which satisfies the boundary condition (REF ). It has well defined momentum \(k_x\)  and frequency \(\omega = \omega _+(k_x,\kappa )\) . Such a state is uniquely defined up to multiples, and for any self-adjoint boundary condition, the quantity \(S(k_x,\kappa ) \in U(1)\)  is called the scattering amplitude, see [1]}. It depends on the choice of bulk sections and on the boundary condition, so we shall rather write \(S(k_x,\kappa ,a)\) .
